Approximation property of Banach spaces

Suppose $T$ is a compact operator on a Hilbert space $V$ and $e_{1}, e_{2}, \ldots$ is an orthonormal basis of range $T$. Let $P_{n}$ denote the orthogonal projection of $V$ onto $\operatorname {span}\left\{ e_{1}, \ldots , e_{n}\right\}$.

Let $H$ be a Hilbert space and $A \in \mathcal{L}(H)$. Prove that the operator $A$ is compact if and only if there exists a sequence of finite-dimensional linear operators which converges uniformly to $A$.

Let $X$ be a Banach space, $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }(X)$, and $A_{n} \xrightarrow {s} I, n \rightarrow \infty$. Prove that a set $M \subset X$ is precompact if and only if the following conditions are fulfilled:

(a) $M$ is a bounded set;

(b) $A_{n} x \rightarrow x, n \rightarrow \infty$, uniformly in $x \in M$.

Verify that condition (b) can be replaced by condition (b') $\forall \varepsilon > 0 \exists n \in \mathbb {N} \forall x \in M:\left\| A_{n} x-x\right\| < \varepsilon$, and condition $\left\{ A_{n}: n \geq 1\right\} \subset S_{\infty }(X)$ can be replaced by condition

Prove that any Banach space in which there is a countable Schauder basis has the approximation property. Prove that any (not necessarily separable) Hilbert space has the approximation property.

Suppose that a Schauder basis exists in a Banach space $X$. Prove that the closure of the subspace of finite-dimensional operators in the sense of strong operator convergence in $\mathcal{B}(X)$ coincides with $\mathcal{B}(X)$.

If $X$ is locally compact and $A \in \mathcal{B}_0(C_0(X))$, then there is a sequence $\left\{ A_n\right\}$ of finite-rank operators such that $\left\| A_n - A\right\| \to 0$.

Let $\mathscr {X}$ be a Banach space and suppose there is a net $\left\{ F_i\right\}$ of finite-rank operators on $\mathscr {X}$ such that (a) $\sup_i \left\| F_i\right\| < \infty$; (b) $\left\| F_i x - x\right\| \to 0$ for all $x$ in $\mathscr {X}$. Show that if $A \in \mathcal{B}_0(\mathscr {X})$, then $\left\| F_i A - A\right\| \to 0$ and hence there is a sequence $\left\{ A_n\right\}$ of finite-rank operators on $\mathscr {X}$ such that $\left\| A_n - A\right\| \to 0$.

Let $1 \le p \le \infty$ and let $(X, \Omega , \mu )$ be a $\sigma$-finite measure space. If $A \in \mathcal{B}_0(L^p(\mu ))$, show that there is a sequence $\left\{ A_n\right\}$ of finite-rank operators such that $\left\| A_n - A\right\| \to 0$.

Let $X$ be compact and let $\mathscr {U}$ be the collection of all pairs $(C, F)$ where $C = \left\{ U_1, \ldots , U_n\right\}$ is a finite open cover of $X$ and $F = \left\{ x_1, \ldots , x_n\right\} \subseteq X$ such that $x_j \in U_j$ for $1 \le j \le n$. If $(C_1, F_1)$ and $(C_2, F_2) \in \mathscr {U}$, define $(C_1, F_1) \le (C_2, F_2)$ to mean:

1. $C_2$ is a refinement of $C_1$; that is, each member of $C_2$ is contained in some member of $C_1$.

2. $F_1 \subseteq F_2$. If $\alpha = (C, F) \in \mathscr {U}$ let $\left\{ \phi_1, \ldots , \phi_n\right\}$ be a partition of unity subordinate to $C$. If $F = \left\{ x_1, \ldots , x_n\right\}$, define $T_\alpha : C(X) \to C(X)$ by

Then:

Every compact operator is the limit (in the norm) of operators of finite rank.

We say that a Banach space $X$ has the approximation property if the identity operator on $X$ can be uniformly approximated on every compact subset of $X$ by operators with finite rank. That is, given a compact set $K \subseteq X$, there must exist continuous finite rank operators $T_{N}$ such that

Let $\mathcal{K}(H)$ denote the set of compact operators in $\mathcal{L}(H)$. Show that $\mathcal{K}(H)$ is a closed subspace of $\mathcal{L}(H)$ that has the set of operators of finite rank as a dense subspace. Is $\mathcal{K}(H)$ an open subset of $\mathcal{L}(H)$ ?